I have seen bubbles made with hydrogen. This is a popular trick with the various lecturers who do fireworks related lectures because the bubbles make a satisfying pop if you ignite them.
A bubble is mainly stabilised by layers of surfactant adsorbed at the gas/water interface. As the bubble wall thins, the adsorbed surfactant layers at the opposite gas/water surfaces come into contact and prevent further thinning. This is a purely kinetic barrier as the gas/water surface tension is still greater than zero (i.e. you'd reduce the overall energy by reducing the surface area) but the rate of desorption of surfactant from the surface is slow.
In principle the gas will affect the adsoption of the surfactant at the gas/water interface and possibly affect the stability, but in practice all common gases are so different from water that the relatively minor differences between gases makes little difference. Bubbles can even be blown with steam as long as you keep the gas phase temperature above 100C.
However, over medium timespans the gas inside the bubble will diffuse out through the water film and cause the bubble to shrink. The rate at which this happens will depend on the solubility of the gas and its diffusion rate in water. I'm sure there will be differences between hydrogen and air, though I don't know of anyone who has actually measured it. I found papers reporting diffusion rates here and here, though both are behind paywalls. I had better luck with solubility figures. Both hydrogen and helium are about a factor of ten less soluble in water than nitrogen, which would make their bubbles more stable than air bubbles though their greater diffusion rates will counteract this to some extent.